1. Field
The present application generally relates to optical metrology, and, more particularly, to the selection of the number of diffraction orders to use in generating a simulated diffraction signal for use in optical metrology measurement, processing, and/or simulation for two-dimensional structures.
2. Related Art
RCWA, differential technique, and similar algorithms have been widely used for the study and design of diffraction structures. The profiles of periodic structures are approximated by a given number of sufficiently thin planar grating slabs. Specifically, RCWA involves three main steps, namely, the Fourier expansion of the field inside the grating; calculation of the eigenvalues and eigenvectors of a constant coefficient matrix that characterizes the diffracted signal; and solution of a linear system deduced from the boundary matching conditions. RCWA divides the problem into three distinct spatial regions: 1) the ambient region supporting the incident plane wave field and a summation over all reflected diffracted orders, 2) the grating structure and underlying unpatterned layers in which the wave field is treated as a superposition of modes associated with each diffracted order, and 3) the substrate containing the transmitted wave field. For a detailed description on the use of RCWA for calculating the simulated diffraction signal off a periodic structure, refer to U.S. Pat. No. 6,891,626, titled CACHING OF INTRA-LAYER CALCULATIONS FOR RAPID RIGOROUS COUPLE-WAVE ANALYSES, by Niu et al., issued on May 10, 2005, incorporated herein by reference in its entirety.
The accuracy of the RCWA solution depends, in part, on the number of terms retained in the space-harmonic expansion of the wave fields, with conservation of energy being satisfied in general. The number of terms retained is a function of the number of diffraction orders considered during the calculations. Efficient generation of a simulated diffraction signal for a given hypothetical profile involves selection of the optimal set of diffraction orders at each wavelength for both transverse-magnetic (TM) and/or transverse-electric (TE) components of the diffraction signal. Mathematically, the more diffraction orders are selected, the more accurate the simulations. However, the higher the number of diffraction orders, the more computation is required for calculating the simulated diffraction signal. Moreover, the computation time is a nonlinear function of the number of orders used. Thus, it is useful to minimize the number of diffraction orders simulated at each wavelength. However, one cannot arbitrarily minimize the number of diffraction orders as this might result in loss of information. The importance of selecting the appropriate number of diffraction orders increases significantly when two-dimensional structures are considered in comparison to one-dimensional structures. Since the selection of the number of diffraction orders is application specific, efficient approaches for selecting the number of diffraction orders is desirable.